Question

Evaluate the following :
$\underset{x\to \frac{1}{\sqrt{2}}}{lim}\frac{x-\cos \left({\sin }^{-1}x\right)}{1-\tan \left({\sin }^{-1}x\right)}$

• A. $\frac{1}{\sqrt{2}}$
• B. $-\frac{1}{\sqrt{2}}$
• C. $\sqrt{2}$
• D. $-\sqrt{2}$

Answer 1

The correct option is B $-\frac{1}{\sqrt{2}}$

Given: $\underset{x\to \frac{1}{\sqrt{2}}}{lim}\frac{x-\cos \left({\sin }^{-1}x\right)}{1-\tan \left({\sin }^{-1}x\right)}$

Let $L=\underset{x\to \frac{1}{\sqrt{2}}}{lim}\frac{x-\cos \left({\sin }^{-1}x\right)}{1-\tan \left({\sin }^{-1}x\right)}$

Substitute $t={\sin }^{-1}x\Rightarrow x=\sin t$

$x\to \frac{1}{\sqrt{2}}\Rightarrow t\to \frac{\pi }{4}$

$\Rightarrow L=\underset{t\to \frac{\pi }{4}}{lim}\frac{\sin t-\cos t}{1-\tan t}$

$=\underset{t\to \frac{\pi }{4}}{lim}\frac{\sin t-\cos t}{1-\frac{\sin t}{\cos t}}$

$=\underset{t\to \frac{\pi }{4}}{lim}\frac{\sin t-\cos t}{\frac{\cos t-\sin t}{\cos t}}$

$=\underset{t\to \frac{\pi }{4}}{lim}\frac{-(\cos t-\sin t)}{\cos t-\sin t}\times \cos t$

$=\underset{t\to \frac{\pi }{4}}{lim}-\cos t=-\frac{1}{\sqrt{2}}$